A NNALES DE L ’I. H. P., SECTION A
R. F. A LVAREZ -E STRADA
The polaron model revisited : rigorous construction of the dressed electron
Annales de l’I. H. P., section A, tome 31, n
o2 (1979), p. 141-168
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141
The polaron model revisited:
rigorous construction of the dressed electron
R. F. ALVAREZ-ESTRADA
Departamento de Fisica Teorica, Facultad de Ciencias Fisicas,
Universidad Complutense, Madrid-3.
Vol. XXXI, n° 2, 1979 Physique theorique.
RESUME. 2014 Le modele du
polar on (c’est-a-dire,
un electron non relativisteen interaction avec un
champ quantifie
dephonons optiques)
est etudiedu
point
de vuemathematique.
Deux methodes sont utilisees pour construirerigoureusement
1’etat d’un electron habille lepolaron).
Lapremiere
est basee sur une combinaison de transformationsd’habillement,
la theorie de
Brillouin-Wigner
et la methode desapproximations
succes-sives : elle est valable seulement pour des valeurs assez
petites
de la constantede
couplage.
La deuxieme methode est basee sur :i)
une version modifieeet
rigoureuse
de la theorie deTamm-Dancoff, specialement
modifiee pour resoudre lesproblemes poses
par le modele dupolaron
pour desgrandes energies
desphonons, ii)
des methodesiteratives,
de Fredholm et de la theorie de ladispersion
pourplusieurs particules,
combinees avec destheoremes de
point
fixe. II est demontre que cette deuxieme methode est valable pour des valeurs croissantes de la constante decouplage,
pour les-quelles
lapremiere
methode n’estplus valable,
et que, d’autre part, elle fournit des fondaments pour faire desanalyses
nonperturbatives.
ABSTRACT. - The
polaron model,
which describes a non-relativistic quantumelectron, interacting
with aquantized optical phonon field,
isstudied from a mathematical
standpoint.
Two methods are used in orderto construct
rigorously
the dressed one-electron state(the polaron).
Thefirst combines a
dressing transformation,
theBrillouin-Wigner approach
and the
contraction-mapping principle,
and worksonly
for rather small values of thecoupling
constant. The second method is based upon :i)
amodified and
rigorous
version of the Tamm-Dancoffapproach, specially adapted
to cope with thepeculiarities
of thepolaron
model atlarge phonon
Annales de l’Institut Henri Poincaré-Section A-Vol. XXXI, 0020-2339/1979/141/$ 4,00/
energies, ii) iterative,
Fredholm andmultiparticle scattering techniques, together withfixed-point
theorems. The secondapproach
can be shownto work for
increasing
values of thecoupling
constant, when the first method breaksdown,
and toprovide
a mathematical basis for non-pertur- bative studies.1. INTRODUCTION
The
large polaron
model describes a slow conduction electroninteracting
with a
quantized optical phonon
field in an ioniccrystal.
References[7-9] ] provide comprehensive
accounts of the differentapproaches,
results and openproblems
in thesubject.
Thepolaron
model isinteresting for,
atleast,
the
following
reasons :i)
it describesphysical phenomena
in threedimen- sional space,ii)
it is free ofdivergences,
at the level ofperturbation theory,
in the one electron
subspace, iii)
it constitutes an excellenttesting ground
for
estimating
the convergence ofperturbative
methods anddeveloping non-perturbative techniques.
There is a vast literature about field-theoretic models related more or less to thepolaron
one, from thestandpoint
ofMathematical
Physics (see [10-?7]).
However, to the author’s
knowledge
and with theexception
of a rathershort discussion
by
Ginibre in[2~],
norigorous study
of thepolaron
modelseems have been
reported
so far. It turns out that thepolaron
model ischaracterized
by
aphonon
energy and a cut-off or vertex functionv(k) having
apeculiar dependence
on the threemomentum k(see
section2),
which
requires
aspecial
treatment.The purposes of this paper are :
1)
to treatrigorously
a class of field- theoretic models which includes the standardlarge-polaron
one,and,
very
specially, 2)
to constructmathematically
the dressed or renormalized one-electron state. The methods to be used and thecorresponding
mainresults are the
following.
a)
Thedressing
transformation usedby
Gross[2~] and
Nelson[2~] will
be combined with the
Brillouin-Wigner
method(see,
forinstance, [29 ])
and the
contraction-mapping principle (section
3 andAppendices A, B).
The main result will be the
rigorous
determination of the dressed electron state via convergentiteration-perturbation methods,
for rather restrictedvalues of the
coupling
constant.b)
A new method will bepresented
in sections4,
5 andAppendix C,
which avoids the use ofdressing
transformations. Itsstarting point
is avariant of the Tamm-Dancoff
approach [30-31 ], specially adapted
to cope with the energy and vertex functioncharacterizing
the modelsunder consideration
(section 4).
One of the main results is therigorous
construction of the dressed electron state for a limited range of values for
Annales de l’Institut Henri Poincaré - Section A
the
coupling
constant(subsection
5.A), through
a new convergent pertur- bation-iteration method which is different from the onepresented
beforein section 3. In the case of the standard
large polaron model,
a numericalanalysis
indicates that convergence now holds for values of thecoupling
constant which may be one to two orders of
magnitude larger
than the upper limit obtained in section 3 for convergenceusing dressing
transformations.Another main result
(subsections
5. Band 5.C)
is therigorous
determination of the dressed electron state forincreasing
values of thecoupling
constant(actually larger
than the onespreviously
considered in subsection5.A), by combining perturbation-iteration,
Fredholm andmulti-particle
scat-tering
methods. This allows for thepossibility
ofexploring,
in asystematic
and
mathematically
controllable way, thenon-perturbative regime
ofvalues for the
coupling
constant. A ratherqualitative
numericalestimate,
which illustrates the last statement, is also
presented.
2. CHARACTERIZATION OF MODELS
We consider a non-relativistic
spinless
quantum electron with baremass mo and
position
and threemomentum operators x =(~), p
=( p~),
=
1, 2,
3( [xl, pj = i~~~),
and an indefinite number oflongitudinal optical phonons, regarded
as scalar bosons. LetI 0 >
be thephonon
vacuumand
a(k), a + (k)
be the standard destruction and creation operators fora
phonon
with(continuously-varying) threemomentum k
and energy( [a(k), ~(~)] = ~~3~(k - 1~), k
=!~!).
The bare one-electron state withthreemomentum q
is denotedby
so that the set of all bare electron-phonon
statesconstitutes a basis for the actual Hilbert space.
By assumption,
the total hamiltonian is :The total threemomentum operator is
, f
andv(k)
are,respectively,
a realcoupling
constant and acomplex
cut-oif function.They satisfy
thefollowing assumptions :
1)
Vol. XXXI, n° 2-1979.
2 lim
(~)
) ~~ ~T7~
either vanishes or is a finite constant,4)
ex!
--+
v° ,
v0being
anon-vanishing complex
constant.The above
assumptions 1)-4)
define aslight generalization
of thephysi- cally interesting large polaron model,
characterizedby
> 0 andv ( ) k - ~ k
for anyphonon threemomentum k, -
whichclearly
satisfiesthem. We shall
apply
all our latterdevelopments
to thelarge polaron model
with
~’
= 20142014_20142014’ Z7T2014 ~
, abeing
the usual dimensionlesscoupling
mo
constant
[32 ].
03C02
Let be the
subspace
of allkets 03C8
such that(Ptot -
= 0 with.2014
~’‘~o~ in order to avoid unwanted « Cerenkov effects »[8 ].
-~0 n
The set of all bare states
kn), q03C0 = 03C0 - > 03A3kj,
constitutesa
complete
orthonormal set inH03C0
with respect to the restricted scalarproduct :
In
Eq. (2.4), ~
denotes the usual sum over the n !permutations
. v( 1 ),...,v(n)
(v(1), ... , v(n)) of (1, ... , n).
Notice that the
ordinary
scalarproduct
of the same states in the fullHilbert space
equals
the restricted one,given
inEq. (2.4),
times a volumedivergent factor, namely, ~~(6).
In the restricted norms of a
state 03C8 belonging
to it and an operator A such that are definedrespectively
as[(~, ~,)~ ] 1~2
and
11
= least upper bound of(
variesthrough-
out
the
ground
state of H(the
dressed electron orpolaron)
with
physical
energy E = and small total threemomentum 03C0:Annales de l’Institut Henri Poincare - Section A
Both 03C8 +
and E are finite in each order ofperturbation theory,
asexplicit calculations, patterned
after those in[4] ] [8] ] [33],
show.Explicitly,
for thelarge-polaron
model it has beenconjectured
thatperturbation theory
converges for 0 . 5
(see
D. Pines in[3 ]).
It is notstraightforward
toprove the convergence of the whole
perturbation
seriesfor t/J +, E,
even forvery small a or
f, and,
to author’sknowledge,
no suchproof
has beenpublished
so far.Actually,
the standard arguments(see,
forinstance, [23]) implying
thatHI
is a smallperturbation
ofHo
and the usual convergenceproof
for the Neumannexpansion
of(z - H) -1
1or t/J +,
for very smallf,
all run into difficulties since
12
= oo.Thus,
arigorous
deter-mination
of I/J +,
E and estimates of the values off
for which convergence holds seem desirable.3. CONSTRUCTION OF DRESSED ELECTRON USING DRESSING TRANSFORMATION AND BRILLOUIN-WIGNER APPROACH
We
perform
thedressing
transformationimplemented by
theunitary
operator exp
T,
whereA
being
anon-negative
fixed constant and8(x)
= 1 if x >0, 8(x)
= 0for x0.
A
lengthy calculation,
similar to those in[2~-2J], yields :
Notice that - 00 E’ 0. The usefulness of this type of
transformation,
in order to renormalize a field-theoretic model more
singular
than the onestudied
here,
was established in[24-25 ].
Itspotential
usefulness for thepolaron
model waspointed
outby
Ginibre in[28 ].
The new hamiltonian H’ can be
given
a mathematical sense,by extending
to it the
rigorous analysis
of Nelson[2~] directly. Appendix
A containssome
rigorous
results which will be useful later in this work.After the
dressing
transformation, the dressed electron state isIt
belongs
to and fulfills[H’ - (E - EW+ =
0. We shallimpose
the normalization = 1 and try to construct
c~ +
from~(~c), by regarding
the latter and allkl
...kn)
asunperturbed
kets and+ as
perturbation.
One could construct
(z - H,)-1
and~’+ by using
theperturbation theory
forquadratic
forms and theprojection techniques presented
in[34] ] (see
also[2~]). However,
it is easier to use theBrillouin-Wigner approach [29],
which leadsdirectly
to :Here, I!
is the unit operator,Q,,
is theprojector
upon insideThe basic
polycy
will consist in :i) solving
the linearEq. (3 . 8) regarding
E as a parameter,ii) plugging
theresulting
solution intothe
right-hand-side
ofEq. (3 . 9)
andsolving
for E.We introduce
The series formed
by
all successive iterations ofEq. (3.8)
can be cast into :Majorizing Eqs. (3.13)
and(3.10),
one findswith x 1
- d 3- k and
0Ai being
£given
inEq. (A . 3).
Annales de l’Institut Henri Poincare - Section A
Next,
we consider themapping
M : E ~M(E),
whereM(E)
isgiven by
the
right-hand-side
ofEq. (3.9),
as well as two valuesEi,
= 1, 2, andtheir associates
M(EJ
under themapping.
Somemajorations yield :
One sees
easily
that for anycomplex E, E1, E2
and any realE, E1, E2
smal-ler than
due to the
projector ’0 - Moreover, using
results fromAppendix A,
we
give
a boundfor I) gl(E1, E2) 111f
inAppendix
B(Eq. (B . 2)),
which shows +~ for fixed 039B, under the same conditions for E1, E2.
Then,
forsufficiently
smallf, : a)
there is a domain D in thecomplex
-2 -2
E-plane, containing
20142014
and20142014
+E’,
which maps into itself underM,
b)
1 for E insideD, c) E2)
1 forE1, E2 belonging
to D.
Then,
the contractionmapping principle (see,
forinstance, [35 ])
ensures the existence of a
unique
fixedpoint
E ofM,
E =M(E),
whichbelongs
to D and can be foundby
successive iterations ofEq. (3.9).
More-over, for the fixed
point, ~ +
isgiven by
the convergent series(3.13).
Theanalysis
of Nelson[25 ]
shows that exp( ± T)
are well definedunitary
operators.
Then,
the truepolaron
state isunambiguously given by
Let us summarize some
qualitative
estimates of the convergence condi- tions for the contractionmapping principle
toapply,
in the case of thestandard
large polaron
model(see
section2)
for 03C0 = 0. One has :2-1979.
We have found that the
condition
1 isroughly
fulfilledwhen A = for values of a up to about
0.009,
which is almostthe order of
magnitude
which characterizes semiconductors of type II-VI[6 ].
The condition 11 1 1 is also
essentially
satisfied under the same conditions.This upper limit for convergence, a ~
0.009,
is about one to two orders ofmagnitude
smaller than the onepreviously conjectured (see
D. Pinesin
[3 ]) :
its smallness is due to theA-dependence
of~,1,
xlwhich,
in turn,comes from the
dressing
transformation. The new method to bepresented
in the
following
sections will notrely
ondressing
transformations and will allow one toimprove
the convergenceconditions,
at least inprinciple.
The actual treatment can be
generalized
when an externalhomogeneous magnetic
field lz =(0,0, h), h
>0, along
thex3-axis
is present. LetAh
=(- ~2,0,0),
e, c be the standard vectorpotential (see,
forinstance, [36 ]),
the electron electric
charge
and thevelocity
oflight
in vacuum.Then,
ifthe sustitution
p
~ p - Ah isdone, Eqs. (2.2-3)
remain valid. Noticec
that the two operators = Pj + =
1,
3 conmute with the actualH,
but p2 + does not.Now,
the basic bare states are(r
=0, 1, 2, ... )
= - 201420142014
, 3 =
(~1.~3)
and theH/s being
the standard Hermite .~ ~ .h "
polynomials.
Let be thesubspace
of allstates 03C8
such thatThe set of all bare states
is a
complete
set for with respect to the new restricted scalarproduct :
Annoles de l’Institut Henri Poincare - Section A
Eqs. (3.1-3), (3.5-7)
and the firstEq. (3.4)
remain validprovided
thatp ~
p - while the secondequation (3.4)
should bereplaced by
By using
p -~ as well as the restricted norms for vectors andc
operators and the
quadratic
form(analogous
to F ofAppendix A)
inducedby (3 . 22)
and the newH’,
one can show that theanalogue
of(A. 1) holds,
with the same e1, e2. Let 20142014 in
Eqs. (3. 8-9)
bereplaced respectively 2mo
by
theground
state and the energy of the electron in the externalmagnetic field, namely, 03C8(03C0;
-0), -’
20142014 +(03C03)2 2m0.
After thesesubstitutions,
the Bril-2 c
2mo
louin-Wigner equations
remain valid and determine theground
state~+
and the energy E of the dressed electron in the external
magnetic
field.The
applicability
of the contractionmapping principle
and the convergenceproofs
can be established as we did before in thissection,
when h = 0.The
polaron
model in presence of an externalmagnetic
field has been studiedpreviously by
several authors[~7-~S]:
our brief discussion above tried toprovide
arigorous justification
of those works(at least,
for theground state).
4. MODIFIED TAMM-DANCOFF APPROACH AND A USEFUL BOUND
The
polaron
state~+
can beexpanded
into bare states as :Thus, J 1l2
is the unnormalizedprobability amplitude
forfinding n
Vol. XXXI, n° 2-1979.
phonons
in thepolaron,
and issymmetric
underinterchanges
ofk’s.
Theinterest of
having
factoredout -1 ~2
will beappreciated shortly. Upon replacing
theexpansion (4.1)
into(H - E)~+ -
0 andusing Eqs. (2.2-4),
one finds the basic recurrence relations
(e1/2n
=en| 1/2) :
for n =
0, 1, 2, 3,
..., withb _ 1
= 0. We shall choose the normalizationI 1 l2
= 1 andintroduce,
for later convenience :Then, Eq. (4.3)
for n = 0 and the set of allEqs. (4.3)
for n 1 becomerespectively
Here,
W
is the linear operator definedby
theright-hand-side
of allEqs. (4.3)
.forn=1,2,3,...
We shall introduce
as well as the
L2-norm
ofbn :
Annales de l’Institut Henri Poincare - Section A
(no
confusion should arise between thenotations ~bn~2 and,
and thefollowing
continued fractions :By recalling Eq. (4 . 7),
the firstEq. (4. 2)
andassumptions 1) - 4)
in sec-tion
2,
one seesthat,
at least forsmall and |E|: i)
2n + oo for n1, ii)
Ln ~ 0 as n oo,iii)
there exists some R > 0 such thatZr
> 0 for any r ~ R.For n
R,
one proves thefollowing
bound :The
inequality (4.9)
can beproved
in two alternative ways :1) By integrating
both sides ofEq. (4. 3) over k1
...kn
andusing
one derives the three-term recurrence of
inequalities :
The solution of the recurrence of
inequalities (4.12)
is outlined inAppen-
dix C.
By setting
=0,
~ LmZ’n ~ Zn, II bn ~2
in(C . 3),
one obtains
(4.9).
2)
Consider the set of allEqs. (4. 3)
for n R >0,
cast it into a matrix form similar toEq. (4 . 6), containing
in theinhomogeneous
terminstead of and iterate the
corresponding
matrix system,thereby generating
an infinite series for R.By taking
theL2-norm
Vol. XXXI, n° 2-1979.
of the series for
bn
andmajorizing
as in alternative1),
one finds amajorizing
series
for ~bn~ 2,
which can be summed into the continued fractionZn
in
(4.9),
thatis,
it coincides with the series obtained whenZn
isexpanded
into power series in
all ~r + 1 ~ 2,
r ~ n. Forbrevity,
we omit details.Notice that to have factored
out |en| -1/2
in theprobability amplitudes
has turned out to be crucial for the
validity
of(4. 9).
Infact,
it leads to theinequalities (4 .10-11 )
with Tn’which,
in turn,contain e1 | 1- 1, ( en + 1 1
1inside the
integrals and, hence,
are finite inspite
ofassumption 4)
in sec-tion 2
(and
so on if the derivation of(4. 9) proceeds through
alternative2)).
We have been unable to derive a recurrence or a bound similar to
(4.12)
ox
(4 . 9) respectively
for the fullprobability amplitudes
dueprecisely
toassumption 4)
and the absence offactors en 1- 1
inside certainintegrals analogous
to Tm Tn+ 1, which nowdiverge.
Remarks. -
i)
A Tamm-Dancoffapproach
to the dressed electron state was alsoproposed by
Larsen in aninteresting
paper[39 ]. However,
he did not factorout en ~ -1~2
and he did not present any bound orrigorous study.
ii)
The Tamm-Dancoffapproach
is related to the so-calledN-quantum approximation :
accounts of the latter appear in[40-41 ].
5. CONSTRUCTION OF DRESSED ELECTRON STATE IN MODIFIED TAMM-DANCOFF APPROACH
We shall construct
rigorously
theamplitudes
>_1,
and thepolaron
energy E in several cases, for
successively increasing
valuesof f.
We assume that
Zr
> 0 for any r ~1,
forgiven
small valuesof f,
in a certain domainD 1
for E. The bound(4 . 9)
for r ~ 2together
withII 112
~(which
can beproved similarly,
as in alternatives1 )
or2)
in section
4)
ensure that the series for all1,
obtainedby
successiveiterations of the system
(4 . 6)
converges in 1.Here,
we shall make theE-dependence
ofbn’s
and T’Sexplicit, frequently.
The direct
majoration
ofM I (E) (Eq. (4.5)) gives:
Next,
we shallstudy
themapping
£M1 :
E ~M1(E).
We consider the set of allEqs. (4.3)
for n 1(or
the system(4.6))
andEq. (4.5)
for twoAnnales de l’Institut Henri Poincaré - Section A
values
Ei, E2
such thatZr(Ei)
>0,r ~ 1,
i =1, 2,
and subtract them.By majorizing
as in alternative1)
of section4,
one finds:The solution of the recurrence of
inequalities (5.
A.3) for bn(E2) [ I 2
Vol. XXXI, n° 2-1979.
is also
given
inAppendix
C.By combining (5.
A.2-3), (5.
A.6-7)
and(C. 3)
one gets :
with the convention =
1, t 2 . Z2
if t =1,
2respectively. Z’n
ish=2
given by Eq. (C . 2),
with-+. T~(E2),
-+T~+ I(E2). By looking
atthe ratio of the
(l
+l)-th
term over the l-th one in the series on theright-
hand-side of
Eq. (5.
A.11)
andnoticing that zn
-+ 0 oo, one showsthat such a series converges.
From
(5. A .1), (4 . 7-8), (5.
A.8-11), (5.
A.4)
and(C . 2)
we see that forgiven f
and smallfixed ~ ~ ~,
there is a domainD2
of values for E such that :i)
itcontains 03C02/2m0, ii)
all continued fractionsZn, Zn
arestrictly positive, iii)
it maps into itself underM 1, iv)
ri2 1 for anyE1, E2 belonging
to it.Again,
the contractionmapping principle guarantees
that there existsa
unique
fixedpoint
E =M1(E) lying
insideD2,
which can be foundby iterating Eq. (4. 5).
For this fixedpoint,
allamplitudes 1,
aregiven by
the convergent seriesgenerated through
the iterations of(4.6).
We shall consider the standard
large polaron
model(recall
section2)
for 7T = 0 and
study numerically
the range ofvalidity
of the aboverigorous
construction of the dressed electron state, for
increasing
values of the dimensionlesscoupling
constant a.Now,
~n(n > 1)
can be evaluatedexplicitly. Then,
in whatfollows,
itwill be understood that Ti 1 and
2,
arereplaced respectively by
~1/2
(1 - E)-1~4 . [an/(n -
1 -(wither
= mo =1).
We have carried out several types of numerical estimates :
a)
We have studied thevalidity
ofZr
> 0 for r ~ 1 and real values of E.Let
Ei ~0
be some energy(not necessarily
thehighest one)
such thatZr(E)
> 0 for r ~ 1 and any E ~Ei.
b)
We have studied theinequality (5 . A .1)
withM1(E)
= E and 7r = 0.For this purpose, we have obtained the solutions
E2
ofwhich fulfill
E 2 E 1-
Annales de l’Institut Henri Poincaré - Section A
For a
0.1,
therigorous
construction of the dressed electron state outlined in this subsection can beproved
to converge. We shall not pre- sent detailed numerical results for such a range.Rather,
we shalldiscuss,
in some
detail,
the moreinteresting
case a ~ 0.1.Table I summarizes our numerical results for
E and
We have shown
numerically
thatZr
> 0 for any r ~ 1 ceases to be true for E >E
1 ifoc=0.5,
0 . 6 and 0 . 7.Moreover,
we have seen that theinequality (5 . A .1 ),
withM 1 (E)
= Eand 7T = 0 is satisfied for
E2
E ~È1.
This means thatE2
can beregarded
as a lower bound for the exact
polaron
energy. Otherpolaron
lower bounds have beenobtained, using
differenttechniques,
in[42 ].
We have studied the
validity
of thecontraction-mapping condition ’12
1(recall (5.A.11))
forgiven a
andvalues of E1, E2
in the rangeE2 E1, E2 È1.
It turns out that ri2 1 is fulfilled for a ~ 0.3. For a =0.4,
our estimates indicate that ri2 is close to 1. For a ~ 0.5 we find that ’12 is
appreciably larger
than 1and,
moreover, that it increases as a does.Notice that the
polaron self-energy
obtained from standard pertur- bationtheory
up to second order[8 ], [33 ], EpT,
andFeynman’s
upper bound for it[8-9 ], EF, satisfy : i) È1
>EpT
>E2
for o. 1 ~ a ~0 . 5, ii) EF>EpT> È1> E2
The main conclusion from the above
analysis
is thefollowing.
Therigorous
construction of the dressed electronpresented
in this subsection doescertainly
converge for a ~ 0.3.Quite probably,
it also converges for ri = 0.4. The last statement is motivatedby
ourprevious
numericalfindings
andby
the fact that ourmajorations
and the contractionmapping principle only give
sufficient conditions for convergence. The above conclu- sionsprovide
apartial
answer to theconjecture by
Pines[3] commented
in section 2. Thus the present
method,
which avoids the use ofdressing transformations,
allows one to establish the mathematical existence of thepolaron
for values of 03B1 which characterize semiconductors of type 111- V[6] ]
and which are one and half orders of
magnitude larger
than the upper limit for convergence0.009)
obtained in section 3. It is uncertain(and
hard to establishnumerically)
whether the mathematical construc-tion of this subsection
actually
converges for a > 0.4.Thus,
for a =0. 5,
the
corresponding
value forEi
1 in Table I could allow for convergenceVol. XXXI, n° 2-1979.
to occur :
however,
we remark thatE2
is ratherlarge
and that 112 is appre-ciably larger
than 1 in this case. Unlessimportant
cancellations occurin the formal solutions obtained
by
successiveiterations,
such convergenceseems more and more doubtful as a increases above
0.4,
since the cor-responding
values forÈ1
andE2
and 112 becomelarge.
All this is in agree- ment,essentially,
with theconjectures
formulatedby
Pines[3 ].
5. B. Another
viewpoint
andstudy
of the case R = 2.Throughout
thissubsection,
we shall assumeZr
> 0 for any r ~2,
forgiven f
and a certain domain of values for E.We cast the set of all
equations (4. 3) for n
2 into a matrix form simi- lar to(4 . 4), (4 . 6), namely :
Here, B2
is the column vector formedby all k2),
... ,bn( k ~
1 ... ...(that is,
it is obtainedby dropping b 1 ( k ~ )
inB1),
is a column vectorwhose elements vanish
identically
except the first one, whichequals
j
=1, 2, j ~
i andW2
is thecorresponding
kernel.Throughout
this sub-section,
the bound(4.9)
remains valid for n2,
so that the series for each 2 obtainedby iterating Eq. (5. B .1)
converges inL2-norm.
Upon considering Eq. (4.3)
for n = 1 andreplacing b2 in it by
theright-
hand-side of
Eq. (4.3)
for n =2,
one finds thefollowing
linearintegral equation
forb 1:
First,
we consider the case " of smallf (so
thatZi
1 > 0 aswell)
and 0 presenta
viewpoint
alternative " and 0complementary
to thatadopted
0 in subsec-Annales de l’Institut Henri Poincare - Section A
tion 5 . A.
Now,
we have amapping M 1 (E
-+M 1 (E)) essentially equi-
valent to the one in subsection 5.
A,
thatis, M 1
is definedby
theright-hand-side
ofEq. (4.5),
and the series formedby
all iterations ofEqs. (5. B .1-2).
One has :but a
slightly
differentfor
will be obtained. One finds :The
inequality (5.
B.6)
can beproved by iterating Eq. (5.
B.2), using
and
summing
theresulting geometric
series.Taking L2-norms
in(5 . B . 3), (5.B.6), using (4 . 9)
for n =2,
3 andsolving for ~b1 ~2,
one arrives at thenew bound :
Assumptions 1 )-4)
in section 2imply
the finitenessof y
1 and Onecan obtain another
inequality,
similar to(5.
A10),
with a newpositive
constant
r~2,
whoselengthy expression
will be omitted. As inprevious
cases, the contraction
mapping principle
can beapplied
to themapping M
1when ~ 2 1,
which leads to constructunique
solutions for E =M 1 (E)
and all 1.
As f increases,
the conditionsZ
> 0 and1, r~ 2 1
can beexpected
to break down.
Then,
theinequality (4.9)
would becomemeaningless
for n = 1, and the convergence of the series
for b1 obtained by iterating (4 . 6)
Vol. XXXI, n° 2-1979.
or
( 5 .
B.2)
would not be warranted. In order to solve thisproblem,
we startby noticing
thatk 1 )
is a Hilbert-Schmidt kernelby
virtue ofassumptions 1)-4)
in section 2.Then,
the modified Fredholmtheory [43] ]
leads to : _Annales de l’Institut Henri Poincaré - Section A
According
to Smithies[43 ],
thefollowing properties
are valid :i)
theseries
(5 . B 12), (5 . B 16)
forki)
and A(the
modified first Fredholm minor and Fredholmdeterminant, respectively) always
converge,by
virtueof the bounds
(5 . B 14-15)
and(5.B. 18), and ~N~2
+ ~,ii) by
assu-ming
4 ~0,
theright-hand-side
ofEq. (5 . B .11) gives
theunique
solu-tion of
Eq. (5.B. 2),
which is valid even when the series formedby
all theiterations of Eq. (5.
B.2) diverges.
The bound(5.
B.8)
is alsovalid, provided
that
Nm(k1, k i )
and0394m
bereplaced by N( k 1, k i )
and 0394respectively.
Eqs. (5 . B .1), (5 . B . 3), (5 . B .11)
and theright-hand-side
of(4.5)
definea new
mapping,
also denotedby M 1 :
E -~Mi(E).
As inprevious
cases,the
application
offixed-point
theorems to the actualmapping
allowsto construct solutions for E =
M 1 (E)
and all 1.We shall
apply
thedevelopments
of this subsection to thelarge polaron
model :
a)
A numerical solution of -E2
=zl(E2)’ ~ bl{E2) ~ 112, II ~1(62) 2 being replaced by
theright-hand-side
of(5.
B.8), yields
values for -E2
whichare
slightly
smaller than those for -E2 (Table I) : thus,
for a =0 . 3,
0 . 4 and0.5,
we getrespectively - E2
=0.362, 0.551,
0.795. This leads to,essentially,
the same conclusions as in subsection 5. A : the method based uponEq. (4 . 5)
and the series formedby
the iterations ofEqs. (5 . B .1-2)
does converge if x ~
0 . 3,
its convergencebeing quite plausible
for a =0 . 4,
and uncertain for a > 0.4(increasingly
doubtful as aincreases).
b)
A numericalstudy
for a >0 . 4,
based uponEqs. (4 . 5), (5. B .1)
andFredholm
theory
is morecomplicated.
For this reason, we shall limit ourselves to somequalitative
estimates. We useseparable approxima-
tions for
kl)
of the typewith n == 1 or 2
(mo
= =1).
Even if it is hard to estimate the error ofsuch an
approximation,
we expect the latter to bemoderately adequate
when
ki)
acts upon functions whosek’1-behavior
ressembles that of the exactb 1 ( k i ),
in a limited range above 03B1 == 0 . 4.Here,
one findsWith
these,
we have carried out a numericalanalysis
similar to the oneoutlined in
a)
above. Instead of -E2,
we now find out numerical solu- tions -63 == 0.515, 0.755,
for II =0.4, 0.5, respectively (essentially
thesame for both n = 1 and n =
2),
which aresystematically
a bit smaller than -E2
or -E2.
This fact could beperhaps regarded
as a numericalVol. XXXI, n° 2-1979.
hint of the
reliability
of Fredholmtheory.
We stress thatZr
> 0 for r ~ 2 isalways
fulfilled and believe thatÈ3
is also a lower bound for the truepolaron
energy.Here,
the mainqualitative
conclusion is : the method based uponEq. (4 . 5),
the series of iterations forEq. (5.
B.1)
andEqs. (5.
B11-18)
can be
expected
to converge for 0.4 ~ a ~ 0 . 5(while
that of subsection 5. A and the one based upon the iterationsof Eq. (5.
B.2)
are of uncertainvalidity
in such a
range),
at least.Here,
we assume thatZr
> 0 if r ~3,
in a certain domain of E and forgiven 7T
and somewhatlarger
valuesof f .
We allow for(4 . 9)
to breakdown for n = 1. 2 so that we do not expect that either
b 1
orb2
could befound
by
successive iterations. The set of allEqs. (4.3) for n
3 can becast into the matrix system
B 3
== +B 3 (B3 being
a column vectorobtained from
Eq. (4 . 4), by dropping bi
1 andb2, etc.) : by iterating
the
latter,
one obtains a series for3,
which converges inL2-norm,
as the bound
(4.9)
ismeaningful
for n 3.We shall concentrate in
displaying
andsolving
a newdifficulty regard- ing b2,
which did not appear when R = 2,omitting
unnecessary details.By considering Eq. (4.3)
for n = 2 andreplacing
in itb3 by
theright-
hand-side of
Eq. (4. 3)
for n =3,
one finds the linearintegral equation :
where j
=1, 2, j
~ i inEqs. (5 .
C .2-3)
and 62 isgiven
inEq. (5 .
B .5). Sym-
2
bolically,
2 we shall rewriteEq. (5 . C .1)
asb2
= +Ii
i= 1b2.
Thekernel
03A3li
is not Hilbert-Schmidt in all threemomentakl, k2, ki, k2,
i= 1
due to the structure of the
right-hand-side ofEq. (5. C.1),
so that the modifiedAnnales de l’Institut Henri Poincare - Section A
Fredholm
theory
cannot be used to solve(5.
as it stands. To solve thisdifficulty,
we shallapply
rearrangementtechniques typical
of multi-particle scattering theory [44 ]. Using symbolic
notationpartially,
the basicnew
equations
read :Notice that either
by iterating
eqs.(5 .
C .5)
and the secondEq. (5 .
C .4)
and
inserting
theresulting
series into the firstEq. (5 .
C .4),
orby iterating directly Eq. (5.
C.1),
one arrives at the same formal series. The main pro-perties
ofEqs. (5 .
C .4-5)
are:a)
The kernelli
is Hilbert-Schmidt inkz, k’i,
for fixed i.Then,
the solution of
Eq. (5 .
C .5)
isgiven by
the modified Fredholmtheory,
that
is, by performing
suitablereplacements
inEqs. (5 .
B.11-17). Sym- bolically,
we writte:03B6i
-t. Z + 1 0394, N.l. L i7
i =1
, 2. Notice that0.
Idepends
1B.
lonly
on i .b)
We notice that the kernel i #j,
is defined for any cp =k2)
as follows:
and so on for
ç 2. ç 1.
In whatfollows,
we shall assume thatand so on
for ~~ 12~k1~’~ !!2. 12,1.
Noticethat 1~,2
+00, [2,1
i + oo. Wecan prove that
- l.
+is
a bounded operator.Vol. XXXI, n° 2-1979.
In
fact, by using
theanalogues
of(5 . B .12)
and(5 . B .14),
some directmajorations give :
The series in
(5.
C.6)
converge, and theproof
for theremaining
contri-butions to is similar.
Moreover,
sinceone finds
easily
a finite bound forII ~
+112
in terms ofII bl 112 and ~b4~2.
c)
The kernel is Hilbert-Schmidt :and
similarly
forÇ2Çl.
Let us sketch a directproof. By using
theanalogues
+00
of
(5.
B.12), namely, N, = ~ +
i == 1,2,
one finds:Annales de l’Institut Henri Poincare - Section A